Lab Astrophysics - University of St Andrews

1

Laboratory astrophysics: Investigation of planetary and astrophysical maser emission

R. Bingham1,2

, D.C. Speirs2, B.J. Kellett

1, I. Vorgul

3, S.L. McConville

2,

R.A. Cairns3, A.W. Cross

2, A.D.R. Phelps

2 and K. Ronald

2

1Central Laser Facility, STFC Rutherford Appleton Laboratory, Chilton, OX11

0QX, U.K.

2Department of Physics, SUPA, University of Strathclyde, Glasgow, G4 0NG,

U.K.

3School of Mathematics and Statistics, University of St Andrews, St Andrews,

KY16 9SS, U.K.

Abstract This paper describes a model for cyclotron maser emission applicable to planetary

auroral radio emission, the stars UV Ceti and CU Virginus, blazar jets and astrophysical shocks.

These emissions may be attributed to energetic electrons moving into convergent magnetic fields

that are typically found in association with dipole like planetary magnetospheres or shocks. It is

found that magnetic compression leads to the formation of a velocity distribution having a

horseshoe shape as a result of conservation of the electron magnetic moment. Under certain

plasma conditions where the local electron plasma frequency ωpe is much less than the cyclotron

frequency ωce the distribution is found to be unstable to maser type radiation emission. We have

established a laboratory-based facility that has verified many of the details of our original

theoretical description and agrees well with numerical simulations. The experiment has

demonstrated that the horseshoe distribution produces cyclotron emission at a frequency just

below the local electron cyclotron frequency, with polarization close to X-mode and propagating

nearly perpendicularly to the electron beam motion. We discuss recent developments in the theory

and simulation of the instability including addressing radiation escape problems, and relate these

to the laboratory, space, and astrophysical observations. The experiments showed strong narrow

band EM emissions at frequencies just below the cold-plasma cyclotron frequency as predicted by

the theory. Measurements of the conversion efficiency, mode and spectral content were in close

agreement with the predictions of numerical simulations undertaken using a particle-in-cell code

and also with satellite observations confirming the horseshoe maser as an important emission

mechanism in geophysical / astrophysical plasmas. In each case we address how the radiation can

escape the plasma without suffering strong absorption at the second harmonic layer.

Keywords Auroral kilometric radiation; cyclotron maser radiation; plasma

instabilities; blazar jets;

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1. Introduction

At least five sources of intense, non-thermal planetary radio emission are known

in our solar system. The Earth, Jupiter, Saturn, Uranus and Neptune all produce

strong maser emission in the kilometre to decametre wavelength range [Zarka,

1992]. In astrophysical environments the cyclotron maser instability is an

important mechanism that provides an explanation for observations of intense

microwave radiation [Treumann 2006]. Radio emission from stars with a dipole

magnetic field UV Ceti [Benz et al.1998, Kellett et al. 2002] and Cu Virginus

[Trigilio et al. 2000, Hatzes 1997, Kellett et al. 2007, Lo et al. 2012], blazar jets

[Begelman et al. 2005, Treumann 2006] and shocks [Bingham et al. 2003] are also

thought to be associated with maser emission. Cyclotron maser emission from

astrophysical objects was first investigated by Twiss [Twiss 1958]. The radiation

is associated with electron beams accelerated into increasing magnetic fields

where the local electron plasma frequency ωpe is much less than the cyclotron

frequency ωce (see figure 1 for the terrestrial auroral case). In the low density

regime, the growth rate of the R-X mode is inversely proportional to the density,

while the growth rate of the ordinary mode is proportional to the density. This

explains why the RX mode is the dominant feature of radiation in regions where

pe << ce [Melrose and Dulk 1982]. In the case of the Earth‟s auroral radio

emission region, the ratio pe / ce ~ 0.01, for blazar jets this ratio can be even

smaller [Begelmann et al. 2005, Treumann 2006]. The idea that planetary radio

emission is the result of a maser instability has been current for many years [Wu

and Lee 1979, Winglee and Pritchett 1986, Melrose 1986, Louarn et al. 1990,

Louarn and Le Queau 1996, Delory et al. 1998, Ergun et al. 2000] with several

schemes having been proposed. The electron loss-cone distribution [Wu and Lee

1979; Winglee and Pritchett 1986] was suggested as the population inversion

required for maser emission. However there is little or no evidence [Delory et al.

1998; Ergun et al. 2000] of the loss -cone distribution in the satellite data to

support this assertion and as pointed out by Melrose (Melrose 1998), for the

emission to be explained by a loss-cone instability it has to be supposed that all

observations are of a saturated state after the loss -cone has filled by pitch angle

diffusion. The loss-cone instability was also questioned as the source of terrestrial

auroral kilometric radiation (AKR) by the results of particle-in-cell simulations

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conducted by Pritchett [Pritchett, 1986; Pritchett et al. 1999], based on observed

electron distribution functions. These simulations show that the loss-cone

instability is not a particularly efficient process. Observations from the Viking

spacecraft [Louarn et al. 1990; Roux et al. 1993] in the Earth‟s auroral

magnetosphere led to the suggestion that the energy source for AKR is instead a

population of downwards accelerated electrons with a large perpendicular velocity

produced by a combination of parallel accelerating electric field and converging

magnetic field lines. The electron distribution that is formed by beam electrons

moving into an increasing magnetic field takes the form of a horseshoe shape.

These earlier observations have been backed up by more recent data from the Fast

Auroral SnapshoT (FAST) satellite [Delory et al. 1998; Ergun et al. 2000;

Pritchett et al. 2002] that clearly demonstrate a strong correlation between the

electromagnetic wave emission and the occurrence of the horseshoe distribution.

A number of authors [Delory et al. 1998; Ergun et al. 2000; Pritchett et al. 2002;

Pritchett et al. 1999; Bingham and Cairns 2000; Bingham et al. 2004; Vorgul et al.

2005; Cairns et al. 2005; Speirs et al. 2005; Ronald et al. 2008; Ronald et al.

2008b; McConville et al. 2008; Speirs et al. 2008; Gillespie et al. 2008] have

recently focused on the horseshoe distribution as the main source of AKR and

demonstrated a robust cyclotron maser instability with a large spatial growth rate

[Bingham and Cairns 2000; Bingham et al. 2004; Vorgul et al 2005], much larger

than for a loss-cone distribution.

FIG. 1. Diagrammatic representation of the terrestrial auroral process.

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Other characteristics that are associated with the horseshoe maser but not the loss-

cone maser are high brightness temperatures > 1014

K, continuous emission and

high efficiency of order 1% [Gurnett 1974]. Given the prevalence of converging

magnetic fields in astrophysics it has been suggested that the horseshoe maser can

explain radio emission from stars with a dipole magnetic field [Benz et al.1998,

Kellett et al. 2002 & 2007, Trigilio et al. 2000, Lo et al. 2012], blazar jets

[Begelman et al. 2005] and shocks [Bingham et al. 2003]. Since the dispersion

relation describing the electromagnetic wave only depends on the factor by which

the magnetic field increases and on ratios of the plasma and cyclotron frequencies,

it is possible to scale the effect to laboratory dimensions. An experiment was

designed and built to test the different cyclotron maser instabilities driven by loss

cone and horseshoe distributions – a difficult task in the space environment. The

experimental apparatus devised consisted of an electron beam injected into the

convergent / fringing field of a magnetic solenoid configuration, creating a

horseshoe distribution in velocity space similar to those observed by spacecraft

[Delory et al. 1998; Ergun et al. 2000; Louarn et al. 1990; Roux et al. 1993;

Pritchett 1986] in the auroral region, Figure 2. An important issue for

consideration is how the radiation can escape from each of the objects discussed

in this paper. For each case we outline possible escape mechanisms.

2. Observations of cyclotron-maser emission

2.1 Satellite observations in the auroral magnetospheric region

The most convenient source for observation is Earth, and terrestrial AKR has been

observed in detail by numerous satellite missions. Such observations from within

the source region indicate that AKR is generated at high altitudes (~1.53 Earth

radii) in cavities of low background plasma density aligned along the auroral

magnetic field. From the first images of the terrestrial auroral electron distribution

function (DE-1 satellite), a definite crescent, or what we call a horseshoe, can be

seen [Menietti and Burch 1985]. Later observations by the Viking and FAST

satellites provided much clearer images where one can see an obvious, highly

populated horseshoe [Delory et al. 1998; Ergun et al. 2000; Pritchett et al. 2002].

The main results of the satellite observations relevant to cyclotron maser emission

are: the existence of fast electrons of approximately the same energy moving into

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a convergent magnetic field, an observed horseshoe shaped electron momentum

distribution, intense kilometric radiation nearly perpendicular to the magnetostatic

field, a well-defined spectral output at around the electron cyclotron frequency, X-

mode polarisation of the emissions within the source region and an estimated

emission efficiency of about 1% [Menietti and Burch 1985; Gurnett 1974; Mutel

et al. 2008; Menietti et al. 2011].

FIG. 2. Illustrative diagram showing the formation of a horseshoe shaped velocity distribution in

an electron beam subject to magnetic compression.

An important issue for consideration is how the radiation can escape from the

auroral magnetospheric plasma into space. The cyclotron maser mechanism

generates radiation propagating almost perpendicular to the magnetic field in the

extraordinary mode. Melrose and Dulk 1982 and Melrose 1999 argue that strong

absorption will take place when the radiation encounters the second harmonic

absorption layer. Second harmonic absorption is maximum when the radiation

propagates perpendicular to the magnetic field and goes to zero as θ approaches

0°, inferring that for radiation propagating at small angles to the magnetic field

attenuation is small [Melrose 1986]. Recent work by Mutel et al 2008 using data

obtained by the four spacecraft Cluster mission show that the radiation from the

auroral region undergoes strong refraction, caused by the density inhomogeneity

in the magnetospheric plasma with altitude [Mutel et al. 2008, Menietti et al.

2011]. Mutel et al. have shown that the ray paths due to refraction are propagating

at angles with respect to the magnetic field of between 10° and 20° well before

they encounter the second harmonic layer, with much reduced attenuation at these

angles. Both simulations [Speirs et al. 2010] and observations [Menietti et al.

2011, Mutel et al. 2008] suggest that the radiation is generated at a backward

angle of a few degrees to the beam direction and not strictly perpendicular. This

serves as a suitable precursor to upward refraction with respect to the decreasing

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background plasma density with increasing altitude as observed by Menietti et al.

2011 and Mutel et al. 2008.

2.2 Stellar observations

Prompted by the first radio resolved image of the star UV Ceti [Benz et al. 1998],

we undertook a review of the x-ray and radio observations of active late-type stars

(in general), and UV Ceti, in particular. One of the aspects of the observations that

we particularly noted was the highly polarized (nearly 100% right-hand circular

polarised along the direction of propagation) radio outbursts of UV Ceti. We also

noted the high degree of correlation observed between the x-ray and radio

luminosity of active late-type stars. Taken together, these observations strongly

suggested that a new radio emission mechanism was responsible for most (and

possibly all) of the radio emission of active late-type stars.

FIG. 3. Diagrammatic representation of radiation emission from the flare star UV-Ceti [Kellett et

al. 2002].

The first resolved image from UV Ceti [Benz et al. 1998] showed that the radio

emission was emerging from the poles of the star. In the analysis by Benz et al.

they estimated that the plasma density was <108 cm

−3 giving a plasma frequency

of <108 Hz, i.e. much less than the observed emission frequency of 8.33 GHz.

Combined with a typical magnetic flux density of order 1000Gauss [Vogt 1980],

we were prompted to develop a cyclotron maser emission model driven by an

electron beam entering the polar converging magnetic region of the star [Kellett et

al. 2002; Bingham et al. 2001]. The fact that this mechanism could easily generate

the observed radio flux from these stars with keV energy electrons could then also

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help explain the x-ray/radio correlation. X-ray emission also implies keV

electrons, so it was possible or even likely that the same electrons were

responsible for the x-ray and radio emission - therefore easily explaining the

observed correlation [Kellett et al. 2002]. Our model for UV Ceti (Bingham et al.

2001) is summarized in Fig. 3. We adopt a dipole field geometry as suggested by

the observations. Similar to the planetary case, a plasma density gradient would

refract the radiation out of the plane perpendicular to the magnetic field resulting

in propagation close to the magnetic field direction before encountering the

second harmonic absorption layer, allowing the radiation to escape with little

absorption. A similar dipole model is also invoked for CU Virginis where we see

pulsed radiation signatures that very much suggest a continuous radiation source

[Kellett et al. 2007, Lo et al. 2012] but which appears pulsed due to the rotation of

the source across the Earth‟s plane. This has been coined by Trigilio the

“lighthouse effect” reminiscent of pulsars but which we could call pulstars

[Trigilio et al. 2007].

We conclude that maser emission from young active stars with a strong dipole-

like magnetic field [Benz et al. 1998, Kellett et al. 2002, Bingham et al. 2001]

may be similar to the planetary auroral models in terms of the strong refraction

induced by altitude variation in magnetospheric plasma density, allowing the

radiation to easily pass through the second harmonic layer with little attenuation.

This may not be so for solar radio emission which is associated with a much more

complex magnetic field structure than stars like UV-Ceti and CU Virginus which

have a strong dipole magnetic field [Benz et al. 1998, Hatzes 1997, Borra and

Landstreet 1980]. It is highly likely that these stars do indeed emit cyclotron

radiation similar to planets from the polar regions of the dipole that correspond to

the auroral regions of planets. In this case we would expect the radiation to be

emitted initially close to perpendicular and refract upwards due to the plasma

density decreasing with altitude. A plasma emission mechanism cannot be ruled

out for stars such as the sun that have more complicated magnetic field structures

[Thejappa G. et al. 2012, Hillan et al. 2012], however stars such as UV-Ceti and

CU-Virginus rotate much faster and have strong dipole magnetic fields of order

1000Gauss [Benz et al. 1998, Hatzes 1997, Borra and Landstreet 1980, Vogt

1980].

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Stellar and planetary cyclotron maser emission is a significant source of

information on planetary / stellar magnetospheres. For offset magnetic / rotational

poles, it can also allow us to study the angular momentum evolution of stars and

planets in order to detect variations in rotational period [Pyper et al. 1998,

Adelman et al. 2001]. The common features of such astrophysical radio sources at

any scale are the presence of a strong dipole or converging magnetic field, the

existence of energetic electrons and observed, highly circularly polarised radio

emissions at a range of frequencies correlating with the source region range in

electron cyclotron frequency [Treumann 2006, Zarka 1992]

2.3 Blazar jets and shocks

Relativistic jets (or blazar jets when directed towards the Earth) are beam-like

linear features observable over a broad range of frequencies and generated

perpendicular to the accretion disc of super massive black holes [Meier et al.

2001, Nemmen et al. 2012]. They can extend over intergalactic distances (many

thousands of light years) and have been observed to generate highly non-thermal

radio emission at frequencies ranging from a few to 100‟s of GHz. It has been

suggested that a cyclotron-maser instability may be responsible for the generation

of these emissions within the low density (ce > pe), magnetised relativistic

electron population of the jet [Bingham et al. 2003, Begelman et al. 2005,

Treumann et al. 2006]. It is believed that small scale magnetic mirrors /

convergent flux tubes may be formed within the jet via hydrodynamic instabilities

or shocks, providing the means of generating the required electron velocity

distribution [Bingham et al. 2003, Begelman et al. 2005].

The magnetic guide fields within a blazar jet are believed to be very strong

[O‟Sullivan and Gabuzda 2009] and for the highly energetic electron population

within the jet this can lead to large relativistic electron cyclotron frequencies in

the 100‟s of GHz range such that ce >> pe [Begelman et al. 2005]. The values

used by Begelman, Ergun and Rees for the ratio of cyclotron frequency to plasma

frequency indicate that the efficiency of the emission process for blazar jets could

be an order of magnitude greater than for the planetary auroral case. This forms

part of the argument used by Begelmann, Ergun and Rees to conclude that the

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radiation could indeed be due to the cyclotron-maser instability driven by an

electron horseshoe / ring distribution. Previous work has shown how a suitable

ring-type distribution may be generated directly by energisation of particles

perpendicular the magnetostatic field via the surfatron mechanism [Katsouleas

and Dawson 1983]. We have also previously proposed a scheme by which

electrons may be accelerated and subject to magnetic compression within a

magnetised collisionless shock, generating a suitable crescent or horseshoe-type

velocity distribution [Bingham et al. 2003]. Both such mechanisms are believed to

be viable within the highly magnetised, turbulent plasma of a blazar jet, with

small-scale magnetic mirrors and field aligned currents expected to occur in

association with quasi-perpendicular collisionless shocks [Begelman et al. 2005].

Counterstreaming ion and ring distributions generated at such shocks can excite

lower hybrid waves via the modified two-stream instability that are capable of

accelerating electrons to high energies, parallel to the magnetic field [McClements

et al. 1993]. These high energy tail distributions can then be subject to magnetic

compression when moving from the upstream to downstream region of the

magnetised collisionless shock, forming a horseshoe-type distribution suitable for

driving the cyclotron-maser instability.

Propagation and escape of cyclotron-maser radiation from a blazar jet has been

considered in some detail [Begelman et al. 2005, Treumann et al. 2006]. Among

the various factors debated, second harmonic cyclotron absorption and

synchrotron absorption are potentially the most significant impediments. These

can however be suitably accounted for [Begelman et al. 2005] in the case of a

blazar jet, where the second harmonic absorption layer is considered to occur

transversely with respect to the magnetostatic field [Begelman et al. 2005] .

Begelmann et al. 2005 have shown as the generated X-mode radiation propagates

radially outwards, where the field can be approximated as dropping off linearly as

B = Bmrm / r with Bm and rm the magnetic flux density and radius at the maser

emission source respectively. Begelman et al. 2005 show that the thickness of the

second harmonic layer can be approximated as ~rmpe / ce which is relatively

thin and results in attenuation of ~10%. Another factor not previously considered

is the potential for refraction of the generated radiation due to the plasma density

gradient associated with quasi-perpendicular shock itself. Akin to our previous

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consideration for the terrestrial auroral case, this could result in R-mode like

radiation propagation and an associated reduction in cyclotron-wave coupling

efficiency for second harmonic absorption in a layer parallel to the magnetic field.

3. Theory and Simulations

3.1 Kinetic Analysis

An analytic form of the electron distribution function with different opening

angles, energy spread and density ratios between the hot component making up

the horseshoe part and a background Maxwellian component is used in a

dispersion relation for the R–X mode which is easily obtainable from the

susceptibility tensor given by Stix [Stix 1992]. We shall assume that the

frequency is close to the electron cyclotron frequency, and also assume that the

Larmor radius is much less than the wavelength for typical electron velocities.

This latter condition means that we need only consider the susceptibility to lowest

order in kv / e. If we neglect all but the zero order terms we get the cold

plasma result. To a first approximation we need only take account of the velocity

distribution of the electrons in the resonant integral which involves 1/( - e)

where e is the relativistic electron cyclotron frequency eB/me with e the

electron charge, B the magnetic field the Lorenz factor and me the electron rest

mass. In this resonant term we must take account of the relativistic shift of the

cyclotron frequency, since this picks out a particular group of resonant electrons

and produces damping or growth of the wave. In terms of momentum p we have

22

2

0

2

1

22

2

02

111

cm

p

cm

peee

(1)

where e0 is the nonrelativistic electron cyclotron frequency. For the real part of

the resonant integral we can simply take the cold plasma value. Although this

goes as 1/( - e0) and appears to be near singular at the resonance, the 1/( -

e0) factors in the real part of the dispersion relation cancel out and it behaves

quite smoothly in the vicinity of the cyclotron frequency. It is not crucial to

include small corrections to the cyclotron frequency in the real part of the

dispersion relation. The refractive index n for the R–X mode is given by

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2

2 xyn

(2)

and considering the dielectric tensor elements and xy with the approximations

we have described, an expression for the perturbation in refractive index due to

the imaginary component can be obtained [Pritchett et al. 2002].

2

0

4

222

0

2

0 )2()(2

ep

peeinn

(3)

Where

0

22

1

1

222

2

0

2

)1(24

1

pp

ee

e

pe f

pp

fpdcm

is represented in spherical polar coordinates (p,µ,) and pp//cos

replaces the usual angle . 21

000 ))(2( eemcp is the resonant momentum.

Substituting the general form of the horseshoe distribution function f0(p,µ) = F(p)

g(µ) into gives

0

2222

2

0

2

24

1

ppe

p

p

FQ

p

FPpcm

where

Term (1) is destabilizing, resulting in emission of waves if the gradient of F is

positive at the resonant momentum. The second term is negative if g is strongly

localized around µ=1 and goes to zero if g becomes uniform on the interval [-1,1].

We might, therefore, have a scenario in which a beam moving down the field line

with small perpendicular spread is stable. As it moves into a region of higher

magnetic field and spreads into a wider horseshoe it can then become unstable and

trigger the emission of AKR.

To illustrate the behaviour we consider the case of a horseshoe distribution

centred at p = 0.1mec, typical of the primary auroral electrons, moving down the

field lines within a lower density Maxwellian plasma with a thermal temperature

of 312eV. Some results suggest that the background Maxwellian may even be

dgP

1

1

21

dgQ

1

1

231(1) (2)

(4)

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absent giving rise to the auroral density cavity. The absence of a low density

background Maxwellian distribution does not change the characteristics of the

instability, in fact it promotes it. We solve Eq. 3 to obtain the perpendicular

spatial growth rate shown in Fig. 4. A typical convective growth length across

the magnetic field Lc = 2/Imk is 10 for µ0 = 0.5, thermal width = 0.02mec and

nh = 0.66ne and 6 for µ0 = 0.75, thermal width = 0.05mec and nh =

0.9ne both for

a cyclotron frequency of 440 kHz. These convective growth distances are 6.8 km

and 4 km, respectively, allowing for many e-foldings within the auroral density

cavity, which has a latitudinal width of about 100 km [Strangeway et al. 1998].

We find that the growth rate decreases for increasing µ0 and increasing thermal

width of the horseshoe distribution and increasing density. The bandwidth is also

extremely narrow, estimated from Fig. 4 to be of order 0.5% or around 200 Hz,

also in agreement with observations.

FIG. 4. Spatial growth rate Im k = normalized to e0 /c as a function of frequency.

In a previous analysis we considered the problem of a ring distribution, unstable

to a cyclotron maser instability, in a plasma with a magnetic field gradient [Cairns

et al. 2008]. This is reminiscent of electron distributions found at shocks and

blazar jets [Bingham et al. 2003, Begelman et al. 2005]. We have shown that it is

possible to have a localized unstable region around the cyclotron resonance with

waves radiating outwards from the beam.

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3.2 PiC simulations

For the purpose of simulating the horseshoe distribution formation and any

subsequent electromagnetic interaction, the 2D axisymmetric version of the finite-

difference time domain PiC (Particle-in-cell) code KARAT was used. KARAT

represents the electric and magnetic fields in a simulated geometry as a

summation of a static component and a time-varying component [Tarakanov

1992]. The time varying electromagnetic fields generated by charges and currents

within the simulation are governed by Maxwell‟s equations (specifically,

Ampere‟s law and Faraday‟s law). The motion of PiC particles within these

electromagnetic fields are governed by the relativistic Lorentz force equation,

with the PiC particle current density at any point within the simulation geometry

determined by the PiC method [Birdsall and Langdon 1985]. The 2D

axisymmetric version of KARAT in particular allows indirect observation of the

distribution of particles in the transverse plane of motion. The code calculates the

PiC electrons radial and azimuthal velocities and plots each particle location in

transverse velocity space. Particles moving in circular orbits transition from v

and vr to -v and -vr periodically. The formation of spatial bunches in a cyclotron

instability corresponds to modulation of electron gyrational velocities (v) and in

the limit of small modulation can be perceived in the 2D plots of v vs vr as

density variations. This is due to the bunch forming in a specific rotational phase

with respect to the local AC field and thus at some instant in time having a

“location” in a plot of v vs vr. In addition, energy modulation can be perceived

through changes in the magnitude of 22

rvvv , illustrated by the area

spanned by PiC particles in plots of v vs vr. A PiC simulation was conducted of

unbounded electron cyclotron-maser emission. The parameters of this simulation

included an axial length (beam propagation path) of 36ce (where ce is the

electron cyclotron free-space wavelength) transverse (radial) dimensions of 5ce

and radiation absorbent boundaries at the radial and axial limits. A grid resolution

of 10 divisions per wavelength at the second harmonic of the electron cyclotron

frequency was used along with a PiC merging factor of 3x106 electrons / PiC

particle. Figure 5 contains a diagrammatic overview with electron beam trajectory

of the unbounded simulation geometry defined within the PiC code KARAT. The

electron beam was injected into the simulation with a predefined horseshoe

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distribution, comprising a pitch spread ||vv of 09.5, beam energy of

20keV±5% and beam current of 14A. A uniform axial magnetic flux density of

0.1T was also used in the unbounded simulation as no magnetic compression was

necessary.

FIG. 5. Diagramatic representation of the PiC simulation geometry, with electromagnetically

absorbent boundaries representative of free space.

PiC particle velocity distributions were plotted after a 200ns run time at two axial

positions within the simulation geometry. The corresponding data is presented in

figure 6. The injected beam distribution at z = 0.36ce shows a well-defined pitch

spread in the v vs vz plot covering the complete pitch range from an axial

electron beam to the point of magnetic mirroring (zero axial velocity). The

corresponding v vs vr plot also shows a uniform spread in relative orbital phase

with no evidence of coherent bunching effects. At z = 11.7ce the picture is very

different however, with clear evidence of azimuthal bunching in the v vs vr plot

[Chu 2004] and in the corresponding v vs vz plot there is spreading in the

transverse velocity profile of high pitch factor electrons. Looking at z = 18ce,

evidence of phase trapping is now present with a concentration of PiC particles

extending to the origin of the v vs vr plot. The corresponding v vs vz plot shows

a saturated state, with minimum vf . All of these characteristics are strongly

Radiation

absorbent

boundaries

15

indicative of resonant particle-wave energy transfer via a cyclotron-maser

instability.

FIG. 6. PiC particle velocity distributions measured on transverse planes within the simulation

geometry at (a) z = 0.36ce (b) z = 11.7ce and (c) z = 18ce.

Figure 7a contains a plot of the radial Poynting flux measured in a plane at r =

0.32ce, over the entire length of the simulation geometry. A DC offset is present

in the measurement due to low frequency electromagnetic field components

associated with the electron beam propagation. The rf output power may therefore

be obtained from the amplitude of the AC signal superimposed on this DC offset.

Looking at figure 7a, there is rather sporadic growth in the electromagnetic output

from 54t/tce onwards (where tce is the electron cyclotron period), with an increased

steady growth observed after 432t/tce saturating at ~2.7kW and corresponding to

an rf conversion efficiency of 0.96%. This is comparable to the 1.3% efficiency

(a)

(b)

(c)

Electron (PiC particle) velocity phase space @ z = 11.7ce

Electron (PiC particle) velocity phase space @ z = 0.36ce

Electron (PiC particle) velocity phase space @ z = 18ce

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obtained from earlier waveguide bounded simulations [Speirs et al. 2008] and

consistent with the generally accepted estimate of ~1% for the AKR generation

efficiency [Gurnett 1974, Pritchett and Strangeway 1985]. The corresponding

output spectra is presented in figure 7b, showing a well-defined spectral

component at 2.68GHz. This represents a 1.1% downshift from the relativistic

electron cyclotron frequency of 2.71GHz.

FIG. 7. (a) Temporal evolution of the radial Poynting flux measured in a plane at r = 0.32ce

spanning the length of the simulation. (b) Fourier transform of Etheta from t/tce = 0 540 at z =

17ce.

Figure 8 contains a 3D contour plot of Etheta mapped over the simulation geometry

after a 200ns run time. An electromagnetic wave sourced at z ~ 13.5ce is evident

propagating near perpendicular to the electron beam with a slight backward wave

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6

Frequency (GHz)

Re

lati

ve

am

pli

tud

e

0

0.005

0.01

0.015

0.02

0.025

0 20 40 60 80 100 120 140 160 180 200

Time (ns)

Po

we

r (M

W)

(b)

(a)

2.7kW

0 54 108 162 216 270 324 378 432 486 540

Electron cyclotron periods

17

character – consistent with the observed 1.1% downshift in the spectral output.

The axial coordinate range over which the wave is generated corresponds to the

position at which a spreading in the PiC particle velocity distribution is first

observed in figure 6, and represents a significant number of Larmour steps from

the point of beam injection for efficient cyclotron-wave coupling to be observed.

This represents a reduction in spatial growth rate compared with earlier

waveguide-bounded simulations [Speirs et al. 2008], although a comparable

saturated RF conversion efficiency was obtained.

FIG. 8. 3D contour plot of Etheta within the simulation geometry. RF emission is evident centred

around an axial position of 18ce and appears to have a slight backward wave character.

4. Laboratory Experiment

An experiment was conducted at the University of Strathclyde‟s beam-plasma

laboratory [Speirs et al. 2005; Ronald et al. 2008; Ronald et al. 2008b;

McConville et al. 2008] to simulate the magnetic compression of an electron

beam and subsequent evolution / stability of the associated electron velocity

distribution. The experimental apparatus was based on the use of an electron gun

to inject particles into an increasing magnetic field produced by a system of

electromagnets. The overall layout of the experiment is illustrated in figure 9,

highlighting the electromagnet configuration formed by six distinct coils. Each

coil was wound from OFHC copper tubing coated with a thin plastic sheath for

0 9

18

36

2

4

27

𝒕

𝒕𝒄𝒆= 𝟓𝟒𝟎

18

electrical insulation. The windings were core cooled by water and were able to

carry a current of up to 450A. The first coil (solenoid 1) was half a metre in length

and formed of two layers, surrounding the electron accelerator and defining the

magnetic flux density experienced by the electrons as they were injected into the

16 cm diameter beam tunnel. Solenoid 2, also half a metre in length and formed of

four layers, confined and transported the electrons to the interaction region and

acted as a transition between the low-field electron gun region and the high-field

interaction region. As the electron beam diameter reduced with increased flux

density, it was possible to reduce the diameter of the interaction region to 8 cm,

which also provided a useful reduction in power consumption of the DC solenoid

arrangement. Solenoids 3–5 provide the maximum plateau flux density (up to 0.5

T) in the apparatus, and were wound as a 10 layer main coil with a pair of two

layer balancing shim coils, respectively. Precise control of the magnetic flux

density in this „interaction‟ region allowed the efficiency of the cyclotron

instability to be investigated as a function of cyclotron resonant detuning.

Electron emission into the apparatus was achieved by placing a cathode electrode

covered with an annular region (3 cm mean radius, 1 cm radial width) of velvet

emitter [Denisov et al. 1998; Speirs et al. 2005; Ronald et al. 2008] within 2 cm of

a sparse mesh anode, figure 9. Design of this injector was undertaken with the aid

of the 2D PiC code KARAT [Speirs et al 2005, 2008]. Application of an

accelerating potential of 75 kV to the cathode led to field-enhanced electron

emission at the discontinuities of the dielectric fibres. The field emission current

was sustained by a tunnelling current and enhanced by an avalanche process

within the valance electrons of the fibres and a surface flashover avalanche along

the outside of the fibres [Noer 1982; Xu and Latham 1992]. A high current

density was emitted from each fibre tip leading to explosive vaporization of the

underlying bulk material resulting in the formation of a cathode plasma flare

[Mesyats 1991] travelling at a velocity of ∼2 cmμs−1

and supporting the space-

charge-limited emission of electrons into the diode gap forming a vacuum spark

discharge.

Most of the current traversed the anode mesh and entered the 16 cm anode can

region of the experiment, propagating into the increasing magnetic field. The

19

electron gun was placed in the fringing field of solenoid 1, figure 9. As the

emission surface was normal to the axis of the solenoids this ensured a variation

in the magnitude of magnetic compression as a function of radial position on the

annular emitter. This induced pitch variation in combination with a 12° domed

electrode in the centre of the emitter annulus ensured the electrons were emitted

with a finite mean pitch factor and pitch spread sufficient to ensure the formation

of a horseshoe shaped velocity distribution when subject to significant magnetic

compression.

The resultant velocity distribution obtained in the experiment had a population

inversion in the perpendicular direction and so would be expected to be unstable

to cyclotron-maser emission, since cyclotron resonance produces diffusion

predominantly in the perpendicular direction in momentum space. The electron

cyclotron resonance condition was defined at the fundamental frequency, with

which we shall be concerned here,

||||vkce

(5)

Here ωce is the electron cyclotron frequency, calculated using the rest mass, and γ

is the usual Lorentz factor. If we consider propagation almost perpendicular to the

steady magnetic field, so that the contribution from the Doppler shift is negligible,

the locus of resonant particles in (p, p) is then just a circle in velocity space. If

this circle lies around the inside of the horseshoe, then we may expect instability

and thus a radiation frequency just below ωce.

20

FIG. 9. Schematic diagram of the experimental setup highlighting the magnetic coil configuration

and convergent axial magnetic field with peak-plateau region for cyclotron resonant energy

transfer.

Near cut-off TE modes were investigated within the interaction waveguide, as

these most closely replicate the propagation (⊥ static B) and polarisation (⊥ static

B) properties of the X-mode, minimising the Doppler broadening of the resonance

[Speirs et al. 2005, Ronald et al. 2008]. Two interaction regimes were

investigated: one comprising a peak (plateau) axial magnetic flux density of 0.18T

for resonance with the TE0,1 mode, and a second with a peak (plateau) axial

magnetic flux density of 0.48T for resonance with the TE0,3 mode. The spectra

illustrated in Figures 10a and 11a were obtained by Fourier transform of a directly

measured AC waveform acquired by a 12GHz single-sequence, digital

oscilloscope. The first (lower) spectral peak in Figure 10a corresponds to the TE0,1

mode being excited at 4.42GHz, whilst the second harmonic may also be observed

at 8.9GHz. The corresponding spectral output for the 11.7GHz resonance regime

is presented in Figure 11a, with a well-defined spectral peak present at 11.7GHz

corresponding to a near cut-off resonance with the TE0,3 mode. On closer

inspection some minor splitting is present in the spectral peak, indicative of some

slight mode competition with the TE2,3 mode.

Mode coupling within the experiment was diagnosed by scanning single mode

rectangular waveguide receivers in the output antenna pattern of the circular

interaction waveguide. Both the 4.42GHz and 11.7GHz interaction regimes were

studied, with figure 10b showing the azimuthally polarised antenna pattern

21

measured for the 4.42GHz resonance experiments. The single cycle pattern

observed, peaking at ~35 degrees is indicative of the TE0,1 mode, and consistent

with expectations from the beam-wave dispersion characteristics for the

experimental parameters [Ronald et al. 2008; Speirs et al. 2008]. Integration over

the radial and azimuthal antenna patterns yielded a total output power of 19kW

corresponding to a beam-wave conversion efficiency of 2% at optimum cyclotron

detuning and a compression ratio of 18. The compression ratio of 9 yielded 35kW

and 1% efficiency, consistent with expectations of the lower compression ratio

yielding a reduction in electron population density at higher pitch factors.

FIG. 10. Experimental measurements for the 4.42GHz resonance, illustrating (a) the spectrum of

the output signal, displaying a strong resonance close to the electron cyclotron frequency,

4.42GHz and (b) the antenna pattern confirming excitation of the near cut-off TE0,1 mode and

yielding the efficiency by integration.

For the 11.7GHz resonance regime, multimode excitation was observed with

some evidence of the TE2,3 mode. The emission was also bursty, with two distinct

22

temporal peaks in the output pulse envelope. Figure 11b shows the corresponding

azimuthal antenna pattern for each of these temporal peaks at a magnetic

compression ratio of 16. The triple peak structure observed is characteristic of

both the TE0,3 and TE2,3 modes. Integrating over the corresponding radial and

azimuthal antenna patterns yielded an output power of 30kW and beam-wave

conversion efficiency of ~1%. Overall, the efficiencies obtained for both the TE0,1

and TE0,3 regimes were in close agreement with the predictions of numerical

simulations and are consistent with efficiency estimates for the generation of

AKR [Gurnett 1974].

FIG. 11. Experimental measurements for the 11.7GHz resonance, illustrating (a) the spectrum of

the output signal, displaying a strong resonance close to the electron cyclotron frequency,

11.7GHz and (b) the antenna patterns confirming excitation of the near cut-off TE0,3 mode and

yielding the efficiency by integration.

23

5. Conclusions

This paper describes an experiment created to reproduce an electron distribution

function similar to those observed to be correlated with the phenomenon of

cyclotron maser radiation commonly found in space and astrophysical plasmas

such as planetary aurora, young active stars with a dipole-like magnetic field,

blazar jets and shocks. The experiment successfully produced the required

electron velocity space distribution. The radiation is associated with electron

beams accelerated downwards into the increasing magnetic dipole field of the

auroral regions where the local electron plasma frequency ωpe is much less than

the cyclotron frequency ωce. Magnetic compression leads to the formation of a

velocity distribution having a horseshoe shape as a result of conservation of

magnetic moment. Good agreement on both the efficiency and spectrum of the

emitted radiation were obtained between experiments and numerical simulations

at 4.45 and 11.7GHz, just below ωce. The evolution of the electron velocity

distribution predicted by the simulations and the narrow bandwidth of the

radiation emissions appears consistent with the expectation of the theoretical

models. Numerical and experimental efficiencies are comparable with the satellite

observations of auroral kilometric radiation. The maser radiation is initially

beamed perpendicular to the magnetic field – an important factor that has bearing

on how the radiation can escape. This has been shown to be possible in the

planetary case by the introduction of a plasma density profile in altitude that

refracts the radiation sufficiently resulting in the beam propagating with small

angles to the magnetic field, reducing the attenuation at the second harmonic layer

[Mutel et al 2008]. We adopted a similar model for the stars UV Ceti and CU

Virginus which both have strong dipole fields. In the case of blaser jets, Begelman

et al. 2005 have demonstrated that second harmonic absorption should be low and

the radiation can escape if the source region is close to the boundary in a

sufficiently thin layer.

Acknowledgements This work was supported by the EPSRC and the STFC Centre for

Fundamental Physics. The authors would like to thank W. He, D. Melrose and V.L. Tarakanov for

helpful discussions.

24

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Figure Legends / Captions

FIG. 1. Diagrammatic representation of the terrestrial auroral process.

FIG. 2. Illustrative diagram showing the formation of a horseshoe shaped velocity

distribution in an electron beam subject to magnetic compression.

FIG. 3. Diagrammatic representation of radiation emission from the flare star UV-

Ceti [3].

FIG. 4. Spatial growth rate Im k = normalized to e0 /c as a function of

frequency.

FIG. 5. Diagramatic representation of the PiC simulation geometry, with

electromagnetically absorbent boundaries representative of free space.

FIG. 6. PiC particle velocity distributions measured on transverse planes within

the simulation geometry at (a) z = 0.36ce (b) z = 11.7ce and (c) z = 18ce.

FIG. 7. (a) Temporal evolution of the radial Poynting flux measured in a plane at r

= 0.32ce spanning the length of the simulation. (b) Fourier transform of Etheta

from t/tce = 0 540 at z = 17ce.

FIG. 8. 3D contour plot of Etheta within the simulation geometry. RF emission is

evident centred around an axial position of 18ce and appears to have a slight

backward wave character.

FIG. 9. Schematic diagram of the experimental setup highlighting the magnetic

coil configuration and convergent axial magnetic field with peak-plateau region

for cyclotron resonant energy transfer.

28

FIG. 10. Experimental measurements for the 4.42GHz resonance, illustrating (a)

the spectrum of the output signal, displaying a strong resonance close to the

electron cyclotron frequency, 4.42GHz and (b) the antenna pattern confirming the

excitation of the near cut-off TE0,1 mode and yielding the efficiency by

integration.

FIG. 11. Experimental measurements for the 11.7GHz resonance, illustrating (a)

the spectrum of the output signal, displaying a strong resonance close to the

electron cyclotron frequency, 11.7GHz and (b) the antenna patterns confirming

the excitation of near cut-off modes and yielding the efficiency by integration.